By means of partial fraction decomposition, the purpose of this paper is to obtain a generalization of an algebraic identity which was given by Chu in \(\textit{The Electronic J. Camb.}\), \(11(2004), \#N15\).

Let \(G\) be a graph on \(n\) vertices \(v_1, v_2, \ldots, v_n\) and let \(d(v_i)\) be the degree of the vertex \(v_i\). If \((d(v_1), d(v_2), \ldots, d(v_n))^t\) is an eigenvector of the \((0,1)\)-adjacency matrix of \(G\), then \(G\) is said to be harmonic. A semi-regular harmonic graph is the harmonic graph which has exactly two different degrees. An equi-bipartite harmonic graph is the bipartite graph \(H = (X, Y; E)\) with \(|X| = |Y|\). In this paper, we characterize the semi-regular harmonic graph and equi-bipartite harmonic graph, and the degree sequence of equi-bipartite \(3\)-harmonic graphs.

We give necessary and sufficient conditions for a resolvable \(4\)-decomposition of \(AK_n\), in the case where \(H\) is one of the 10 graphs obtained by the union of two paths of length 2, with two possible exceptions. In particular, we complete the \(4\)-star (\(\lambda\)) and \(T\) (\(\tau\)) for higher \(\lambda\) and give complete solutions for resolvable decompositions into Fish (\(4\)-\(3\)), Mulinetto (\(hx\)) and Kites (\(BSI\)). In the cases of the Fish and Mulinetto the solution is obtained \(1\)-rotationally.

We note that with only a slight modification, Su’s proof on the fragments in \(k\)-critical \(n\)-connected graphs (see J. Graph Theory \(45 (2004), 281-297\)) can imply the following more general result: every non-complete \(W\)-locally \(k\)-critical \(n\)-connected graph has \(2k + 2\) distinct fragments \(F_1, F_2, \ldots, F_{2k+2}\) such that \(F_1 \cap W, F_2 \cap W, \ldots, F_{2k+2} \cap W\) are pairwise disjoint.

A packing of a graph \(G\) is a set of edge-disjoint \(4\)-cycles in \(G\) and a maximum packing of \(G\) with \(4\)-cycles is a packing which contains the largest number of \(4\)-cycles among all packings of \(G\). In this paper, we obtain the maximum packing of certain graphs such as \(K_{2m+1} – H\) where \(H\) is a \(2\)-regular subgraph, \(K_{2m} – F\) where \(F\) is a spanning odd forest of \(K_{2m}\), and \(2K_{2m} – L\) where \(L\) is a \(2\)-regular subgraph of \(2K_{2m}\).

In this paper, we consider the relationships between the second order linear recurrences, and the generalized doubly stochastic permanents and determinants.

An \(\lambda\)-design on \(v\) points is a set of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(\alpha\)-designs can be obtained from symmetric designs by a complementation procedure. In a previous paper, the author established feasibility criteria for the existence of \(\lambda\)-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. In that paper, these criteria and a computer were used to prove that the \(\lambda\)-design conjecture is true for all \(\lambda\)-designs with two block sizes with \(\lambda \leq 90\) and \(\lambda \neq 45\). In this paper, we extend these results and prove that the \(\lambda\)-design conjecture is also true for all \(\lambda\)-designs with two block sizes with \(\lambda = 45\) or \(91 \leq \alpha < 150\).

The binary linear code \(H^\bot_{m,2}\), \(m > 2\), of length \(\binom{m}{2}\) represented by the generator matrix \(H_{m,2}\) consisting of all distinct column strings of length \(m\) and Hamming weight \(2\) is considered. A parity-check matrix \(H^\bot_{m,2}\) is assigned to the code \(H^\bot_{m,2}\). The code \(H_{m,2,3}\), \(m > 3\), of length \(\binom{m}{2} + \binom{m}{3}\) represented by the parity-check matrix \(H_{m,2,3}\) consisting of all distinct column strings of length \(m\) and Hamming weight two or three is also considered. It is shown that \(H^\bot_{m,2}\) and \(H_{m,2,3}\) are optimal stopping redundancy codes, that is for each of these codes the stopping distance of the associated parity-check matrix is equal to the minimum Hamming distance of the code, and the rows of the parity-check matrix are linearly independent. Explicit formulas determining the number of stopping sets of arbitrary size for these codes are given.

For a finite group \(G\) and subsets \(T_1, T_2\) of \(G\), the Bi-Cayley digraph \(D = (V(D), E(D)) = D(G, T_1, T_2)\) of \(G\) with respect to \(T_1\) and \(T_2\) is defined as the bipartite digraph with vertex set \(V(D) = G \times \{0, 1\}\), and for \(g_1, g_2 \in G\), \(((g_1, 0), (g_2, 1)) \in E(D)\) if and only if \(g_2 = t_1 g_1\) for some \(t_1 \in T_1\), and \(((g_1, 1), (g_2, 0)) \in E(D)\) if and only if \(g_1 = t_2 g_2\) for some \(t_2 \in T_2\). If \(|T_1| = |T_2| = k\), then \(D\) is \(k\)-regular. In this paper, the spectra of Bi-Circulant digraphs are determined. In addition, some asymptotic enumeration theorems for the number of directed spanning trees in Bi-Circulant digraphs are presented.

The genus of a graph \(G\), denoted by \(\gamma(G)\), is the minimum genus of an orientable surface in which the graph can be embedded. In the paper, we use the Joint Tree Model to immerse a graph on the plane and obtain an associated polygon of the graph. Along the way, we construct a genus embedding of the edge disjoint union of \(K\) and \(H\), and solve Michael Stiebitz’s proposed conjecture: Let \(G\) be the edge disjoint union of a complete graph \(K\) and an arbitrary graph \(H\). Let \(H’\) be the graph obtained from \(H\) by contracting the set \(V(X)\) to a single vertex. Then

\[\gamma(K) + \gamma(H’) \leq \gamma(G).\]